We are carrying out a survey to search for giant extrasolar planets around nearby, moderate-age stars in the mid-infrared L' and M bands (3.8 and 4.8 microns, respectively), using the Clio camera with the adaptive optics system on the MMT telescope. To date we have observed 7 stars, of a total 50 planned, including GJ 450 (distance about 8.55pc, age about 1 billion years, no real companions detected), which we use as our example here. We report the methods we use to obtain extremely high contrast imaging in L', and the performance we have obtained. We find that the rotation of a celestial object over time with respect to a telescope tracking it with an altazimuth mount can be a powerful tool for subtracting telescope-related stellar halo artifacts and detecting planets near bright stars. We have carried out a thorough Monte Carlo simulation demonstrating our ability to detect planets as small as 6 Jupiter masses around GJ 450. The division of a science data set into two independent parts, with companions required to be detected on both in order to be recognized as real, played a crucial role in detecting companions in this simulation. We mention also our discovery of a previously unknown faint stellar companion to another of our survey targets, HD 133002. Followup is needed to confirm this as a physical companion, and to determine its physical properties.